Certification Problem

Input (TPDB TRS_Standard/AProVE_04/JFP_Ex51)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  if_active#(false,x,y)	→	mark#(y)	(32)
  mark#(s(x))	→	mark#(x)	(21)
  mark#(div(x,y))	→	mark#(x)	(25)
  mark#(div(x,y))	→	div_active#(mark(x),y)	(26)
  div_active#(s(x),s(y))	→	if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0)	(30)
  if_active#(true,x,y)	→	mark#(x)	(31)
  mark#(if(x,y,z))	→	mark#(x)	(27)
  mark#(if(x,y,z))	→	if_active#(mark(x),y,z)	(28)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [ge_active(x1, x2)]	=	3 · x1 + 0 · x2 + -∞
  [ge(x1, x2)]	=	1 · x1 + 0 · x2 + 2
  [minus_active(x1, x2)]	=	0 · x1 + -∞ · x2 + 4
  [s(x1)]	=	0 · x1 + 1
  [div_active#(x1, x2)]	=	-∞ · x1 + 3 · x2 + -∞
  [if_active#(x1, x2, x3)]	=	-∞ · x1 + 0 · x2 + 0 · x3 + 2
  [true]	=	2
  [div_active(x1, x2)]	=	0 · x1 + 3 · x2 + 0
  [0]	=	4
  [false]	=	0
  [mark#(x1)]	=	0 · x1 + 2
  [if_active(x1, x2, x3)]	=	0 · x1 + -∞ · x2 + -∞ · x3 + -∞
  [if(x1, x2, x3)]	=	0 · x1 + 2 · x2 + 5 · x3 + 3
  [div(x1, x2)]	=	0 · x1 + 3 · x2 + 1
  [minus(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0
  [mark(x1)]	=	-∞ · x1 + 0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  mark#(if(x,y,z))

  →

  if_active#(mark(x),y,z)

  (28)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [ge_active(x1, x2)]	=	0 · x1 + 0 · x2 + 0
  [ge(x1, x2)]	=	0 · x1 + 1 · x2 + 0
  [minus_active(x1, x2)]	=	6 · x1 + 0 · x2 + 0
  [s(x1)]	=	0 · x1 + 0
  [div_active#(x1, x2)]	=	-∞ · x1 + 4 · x2 + 0
  [if_active#(x1, x2, x3)]	=	-∞ · x1 + 0 · x2 + 4 · x3 + 1
  [true]	=	4
  [div_active(x1, x2)]	=	-∞ · x1 + 0 · x2 + -∞
  [0]	=	0
  [false]	=	0
  [mark#(x1)]	=	0 · x1 + -∞
  [if_active(x1, x2, x3)]	=	0 · x1 + 0 · x2 + 7 · x3 + 0
  [if(x1, x2, x3)]	=	3 · x1 + 0 · x2 + 0 · x3 + 0
  [div(x1, x2)]	=	0 · x1 + 4 · x2 + 0
  [minus(x1, x2)]	=	-∞ · x1 + 0 · x2 + 4
  [mark(x1)]	=	0 · x1 + -∞

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

  if_active#(false,x,y)	→	mark#(y)	(32)
  mark#(if(x,y,z))	→	mark#(x)	(27)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [ge_active(x1, x2)]	=	0 · x1 + 0 · x2 + 0
  [ge(x1, x2)]	=	-∞ · x1 + 0 · x2 + 2
  [minus_active(x1, x2)]	=	5 · x1 + 1 · x2 + 0
  [s(x1)]	=	0 · x1 + -∞
  [div_active#(x1, x2)]	=	-∞ · x1 + 1 · x2 + 7
  [if_active#(x1, x2, x3)]	=	-∞ · x1 + 0 · x2 + 7 · x3 + 0
  [true]	=	2
  [div_active(x1, x2)]	=	-∞ · x1 + 0 · x2 + 1
  [0]	=	0
  [false]	=	4
  [mark#(x1)]	=	0 · x1 + -∞
  [if_active(x1, x2, x3)]	=	0 · x1 + 0 · x2 + -∞ · x3 + 0
  [if(x1, x2, x3)]	=	0 · x1 + 0 · x2 + 1 · x3 + 1
  [div(x1, x2)]	=	2 · x1 + 1 · x2 + 7
  [minus(x1, x2)]	=	-∞ · x1 + -∞ · x2 + 4
  [mark(x1)]	=	0 · x1 + 0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

  mark#(div(x,y))

  →

  mark#(x)

  (25)

  could be deleted.

  1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [ge_active(x1, x2)]	=	0 · x1 + -∞ · x2 + 2
  [ge(x1, x2)]	=	0 · x1 + -∞ · x2 + 2
  [minus_active(x1, x2)]	=	-∞ · x1 + -∞ · x2 + 0
  [s(x1)]	=	0 · x1 + 1
  [div_active#(x1, x2)]	=	1 · x1 + -∞ · x2 + 0
  [if_active#(x1, x2, x3)]	=	0 · x1 + 0 · x2 + 1 · x3 + 2
  [true]	=	1
  [div_active(x1, x2)]	=	2 · x1 + -∞ · x2 + 1
  [0]	=	0
  [false]	=	1
  [mark#(x1)]	=	0 · x1 + 0
  [if_active(x1, x2, x3)]	=	0 · x1 + 0 · x2 + 0 · x3 + 3
  [if(x1, x2, x3)]	=	0 · x1 + 0 · x2 + 0 · x3 + 3
  [div(x1, x2)]	=	2 · x1 + -∞ · x2 + 1
  [minus(x1, x2)]	=	-∞ · x1 + -∞ · x2 + 0
  [mark(x1)]	=	0 · x1 + -∞

  together with the usable rules

  minus_active(0,y)	→	0	(1)
  mark(0)	→	0	(2)
  minus_active(s(x),s(y))	→	minus_active(x,y)	(3)
  mark(s(x))	→	s(mark(x))	(4)
  ge_active(x,0)	→	true	(5)
  mark(minus(x,y))	→	minus_active(x,y)	(6)
  ge_active(0,s(y))	→	false	(7)
  mark(ge(x,y))	→	ge_active(x,y)	(8)
  ge_active(s(x),s(y))	→	ge_active(x,y)	(9)
  mark(div(x,y))	→	div_active(mark(x),y)	(10)
  div_active(0,s(y))	→	0	(11)
  mark(if(x,y,z))	→	if_active(mark(x),y,z)	(12)
  div_active(s(x),s(y))	→	if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0)	(13)
  if_active(true,x,y)	→	mark(x)	(14)
  minus_active(x,y)	→	minus(x,y)	(15)
  if_active(false,x,y)	→	mark(y)	(16)
  ge_active(x,y)	→	ge(x,y)	(17)
  if_active(x,y,z)	→	if(x,y,z)	(18)
  div_active(x,y)	→	div(x,y)	(19)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  mark#(div(x,y))

  →

  div_active#(mark(x),y)

  (26)

  could be deleted.

  1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    mark#(s(x))

    →

    mark#(x)

    (21)

    1.1.1.1.1.1.1.1 Size-Change Termination

    Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

    mark#(s(x))	→	mark#(x)	(21)
    1	>	1

    As there is no critical graph in the transitive closure, there are no infinite chains.

• The 2^nd component contains the pair

  ge_active#(s(x),s(y))

  →

  ge_active#(x,y)

  (24)

  1.1.2 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  ge_active#(s(x),s(y))	→	ge_active#(x,y)	(24)
  2	>	2
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 3^rd component contains the pair

  minus_active#(s(x),s(y))

  →

  minus_active#(x,y)

  (20)

  1.1.3 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  minus_active#(s(x),s(y))	→	minus_active#(x,y)	(20)
  2	>	2
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.